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Dynamo action is demonstrated numerically in the forced Taylor–Green (TG) vortex made up of a confined swirling flow composed of a shear layer between two counter-rotating eddies, corresponding to a standard experimental setup in the study of turbulence. The critical magnetic Reynolds number above which the dynamo sets in depends crucially on the fundamental symmetries of the TG vortex. These symmetries can be broken by introducing a scale separation in the flow, or by letting develop a small non-symmetric perturbation which can be either kinetic and magnetic, or only magnetic. The nature of the boundary conditions for the magnetic field (either conducting or insulating) is essential in selecting the fastest growing mode; implications of these results to a planned laboratory experiment are briefly discussed.

A new transient regime in the relaxation towards absolute equilibrium of the conservative and time-reversible 3-D Euler equation with high-wavenumber spectral truncation is characterized. Large-scale dissipative effects, caused by the thermalized modes that spontaneously appear between a transition wavenumber and the maximum wavenumber, are calculated using fluctuation dissipation relations. The large-scale dynamics is found to be similar to that of high-Reynolds number Navier-Stokes equations and thus to obey (at least approximately) Kolmogorov scaling., 4 pages, 5 figures new version with only 4 figures; title changed; manuscript changed; accepted by PRL

Preliminary results on the behavior of a forced pendulum chain modified in order to obtain a tunable frequency gap in the dispersion relation are presented. The limit of vanishing frequency gap is considered. Discrepancy with previous results [Physica D 163 (2002) 127-149] are discussed.

Non-Isotropic Attractive Bose-Einstein condensates are investigated with Newton and inverse Arnoldi methods. The stationary solutions of the Gross-Pitaevskii equation and their linear stability are computed. Bifurcation diagrams are calculated and used to find the condensate decay rates corresponding to macroscopic quantum tunneling, two-three body inelastic collisions and thermally induced collapse. Isotropic and non-isotropic condensates are compared. The effect of anisotropy on the bifurcation diagram and the decay rates is discussed. Spontaneous isotropization of the condensates is found to occur. The influence of isotropization on the decay rates is characterized near the critical point., Comment: revtex4, 11 figures, 2 tables. Submitted to Phys. Rev. A

We consider the generation of a large-scale magnetic field by a turbulent flow driven by a small-scale helical forcing in a low magnetic Prandtl number fluid. We provide an estimate of the dynamo threshold that takes into account the presence of large-scale turbulent fluctuations by considering that the scales of the flow that mostly contribute to the dynamo process are roughly in absolute equilibrium. We show that turbulent flows in absolute equilibrium do generate dynamos and we compare their growth rates to their laminar counterparts. Finally, we show that the back reaction of the growing magnetic field modifies the statistical properties of turbulent flow by suppressing its kinetic helicity at large magnetic Reynolds number.

Dynamo action is demonstrated numerically in the forced Taylor-Green (TG) vortex [1] made up of a confined swirling flow composed of a shear layer between two counter-rotating eddies and corresponding to a standard experimental set-up in the study of turbulence called the von Karman (VK) swirling flow. The critical magnetic Reynolds number above which the dynamo sets in depends crucially on the fundamental symmetries of the TG vortex which can be broken by introducing a scale separation in the flow, or by letting develop a small non-symmetric perturbation which can be either kinetic and magnetic, or only magnetic [2]. We present cases where a long term magnetic field produced by dynamo action saturates. Implications of our results to VK sodium experiments are discussed.

The aim of this short set of lectures is to make available in a self-contained form the basics of classical turbulence theory. A first section is concerned with orders of magnitude. How can one determine what the size of a parachute on a Martian probe should be? The second section is devoted to exact results. It culminates with the demonstration of the four-fifth law obeyed by the third order structure function. Finally, the last section is about intermittency. The multifractal scaling behavior of the pdf of velocity increments and the associated structure functions is investigated. A new double asymptotic relation between pdf and moments is derived.

The nonlinear vorticity dynamics of the Batchelor trailing vortex is presented. Direct numerical simulations of the three-dimensional, incompressible, Navier-Stokes equations by a spectral method are used. It is shown that the initial vortex core is subjected to three transformations: a twisting phase, a lateral expansion of its cross section and formation of a spiral structure. These transformations are accompanied by a gradual deceleration of axial velocity. Mean kinetic energy is then transferred to radial velocity. When the transfer is maximum a secondary instability is initiated leading to the spiral structure. These transformations are in agreement with vortex breakdown observed in recent experiments.

Superfluid turbulence is studied experimentally in a low-temperature Helium flow and numerically with the Gross-Pitaevskii equation in the Taylor-Green (TG) vortex flow. It is known that [1, 2] the kinetic energy transfer is of the same magnitude in both the superfluid and viscous TG vortex. Furthermore, the energy spectrum of the superflow is compatible with Kolmogorov’s scaling. Many vortex reconnection events take place throughout both the superfluid and viscous flow. Experimentally, pressure fluctuation spectra show little difference above and far below the superfluid transition temperature, where the normal-fluid component of Helium is less than 5% in mass (at T = 1.2K).

International audience; Low-temperature superfluid turbulence is studied experimentally in a Helium swirling flow and numerically with the Gross-Pitaevskii equation in the geometry of the Taylor-Green (TG) vortex flow. Numerically, it was found in Nore et al. (1997a, b) that the kinetic energy transfer in the superfluid TG vortex is comparable to that of the viscous TG vortex and that the energy spectrum of the superflow is compatible with Kolmogorov's scaling. The vorticity dynamics of the superflow are similar to that of the viscous flow. In both cases, many vortex reconnection events happen throughout the flow. Experimentally, power measurements and pressure fluctuation spectra show very little difference above and far below the superfluid transition temperature, where the normal-fluid component of Helium is negligible (less than 5% in mass at T = 1.2 K).

Low-temperature decaying superfluid turbulence is studied using the nonlinear Schr\"odinger equation in the geometry of the Taylor-Green (TG) vortex flow with resolutions up to ${512}^{3}$. The rate of (irreversible) kinetic energy transfer in the superfluid TG vortex is found to be comparable to that of the viscous TG vortex. At the moment of maximum dissipation, the energy spectrum of the superflow has an inertial range compatible with Kolmogorov's scaling. Physical-space visualizations show that the vorticity dynamics of the superflow is similar to that of the viscous flow, including vortex reconnection. The implications to experiments in low-temperature helium are discussed.

The connexion of nonlinear wave equations with the dynamics of barotropic fluids by Madelung’s transformation is reviewed in the case of fluids with arbitrary equations of state. Numerical simulations of the Nonlinear Schrodinger Equation (NLSE) reproducing the instabilities of non rotating and rotating cylindrical jets are presented. It is shown that NLSE, a dynamical model of superflows, reproduces many flow features usually obtained in the context of Euler or Navier-Stokes equations.

The dynamics of three-dimensional ideal flows is investigated by direct numerical simulations of the Euler equations at resolutions up to 2563 for general periodic flows and 8643 for the symmetric Taylor-Green vortex. The spontaneous emergence of flat pancake-like structures that shrink exponentially in time is observed. A simple self- similar model that fits these observations is presented. Focusing instabilities similar to those leading to streamwise vortices in the context of free shear layers (15), are expected to subsequently concentrate the vorticity and produce isolated vortex filaments. A finite time singularity for the Euler equations is not excluded as the result of interactions among these filaments.

The generation of side jets from axisymmetric jets is studied by direct numerical integration of the Navier-Stokes equations. After vortex rings are formed via the primary Kelvin-Helmholtz instability, pairs of axially counterrotating vorticity filaments appear which eject fluid out of the jet. We describe the three-dimensional topology of the rings, filaments, and side jets.

The three-dimensional Navier-Stokes equations are numerically integrated by a spectral method. Using the symmetries of the initial data a resolution of 8643 and high Reynolds numbers (RA = 140) are obtained. Visualisations of the resulting turbulent flow show that the vorticity is spatially more concentrated than the energy dissipation. Consequently the turbulent activity is strongly correlated with low pressure zones. Measures of the exponent of fractal codimension confirm these results by showing more intermittency for vorticity: u = 0.86, than for energy dissipation: fl = 0.38.

New generalized equations of motion for the Weber-Clebsch potentials that describe both the Navier-Stokes and MHD dynamics are derived. These depend on a new parameter, which has dimensions of time for Navier-Stokes and inverse velocity for MHD. Direct numerical simulations are performed. For Navier-Stokes, the generalized formalism captures the intense reconnection of vortices of the Boratav, Pelz and Zabusky flow, in agreement with the previous study by Ohkitani and Constantin. For MHD, the new formalism is used to detect magnetic reconnection in several flows: the 3D Arnold, Beltrami and Childress (ABC) flow and the (2D and 3D) Orszag-Tang vortex. It is concluded that periods of intense activity in the magnetic enstrophy are correlated with periods of increasingly frequent resettings. Finally, the positive correlation between the sharpness of the increase in resetting frequency and the spatial localization of the reconnection region is discussed.

Generalized equations of motion for the Weber-Clebsch potentials that reproduce Navier-Stokes dynamics are derived. These depend on a new parameter, with the dimension of time, and reduce to the Ohkitani and Constantin equations in the singular special case where the new parameter vanishes. Let us recall that Ohkitani and Constantin found that the diffusive Lagrangian map became noninvertible under time evolution and required resetting for its calculation. They proposed that high frequency of resetting was a diagnostic for vortex reconnection. Direct numerical simulations are performed. The Navier-Stokes dynamics is well reproduced at small enough Reynolds number without resetting. Computation at higher Reynolds numbers is achieved by performing resettings. The interval between successive resettings is found to abruptly increase when the new parameter is varied from 0 to a value much smaller than the resetting interval.

The simplest interaction between sound and quantized superfluid vortices is investigated both analytically in the Born approximation and numerically, using the nonlinear Schr\"odinger equation. Compared with the scattering by a classical vortex, the quantum result reflects the structure of the vortex core and has an extra phase shift. The order of magnitude of the effects suggests possible experiments in rotating superfluid helium.

The interaction of a classical field with general tensorial character with a particle described by its trajectory is shown to lead in general to a Lorentz-Dirac-type equation, independently of the details of the interaction. The divergences of the theory are carefully analysed by a method we have proposed before, which is shown to be equivalent to other regularization schemes that have been used previously. We apply our results to several well-known interactions including linearized gravitation and symmetric electrodynamics with monopoles.

We here report results obtained from numerical simulations of the TaylorGreen three-dimensional vortex flow. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier-Stokes equations (with up to 2563 modes) and by power series analysis in time. The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation. The flow is smooth but has complex-space singularities within a distanceδ(t) of the real space which are manifest through an exponential tail in the energy spectrum. It is found thatδ(t) decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place at later times, possibly leading to a real singularity (δ=0) at a finite time. These direct integration results are consistent with new presented results extending the Morf, Orszag, and Frisch temporal power series analysis from order t40 to order t80. Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis. The viscous dynamics (decay) have been studied for Reynolds numbersR (based on integral scale) up to 3000 and beyond the time tmax at which the maximum energy dissipation is achieved. Early time, highR dynamics are essentially inviscid and laminar. Then, instabilities starting at small scales, which may be driven by viscosity, make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Neart max the small scales of the flow are nearly isotropic providedR>1000. Various features characteristic of fully developed turbulence are observed neart max whenR=3000.

We derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems. Using an appropriate scalar product in the space of homogeneous vector polynomials, we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator. This leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form. Explicit examples, and results obtained when there are additional symmetries, are also presented.

We construct functional integral and operator formalisms for the stochastic process generated by a modified Langevin equation which contains a white noise and an independent markovian process taking discrete values.

We study the spatial stability of systems at onset of the temporal intermittent transition to chaos. We show that the phase dynamics during laminar episodes, described as orbits near a saddle-node bifurcation of limit cycles, is second order in time and can lead to propagative behaviour.

Approximation procedures for cubic stochastic processes are tested, in particular for their realisability properties. This is done on the algebraic cubic equation pu(t)+ Av3(r) =f(t) with zero-mean stationary gaussian drivingf(t). This equation is a limiting case for slowly varying driving of the Van der Pol, the Duffing and the cubic nonlinear Langevin equations. Exact solutions are compared with various field-theoretic approximation pro- cedures generated by a variant of the Martin-Siggia-Rose formalism adapted to algebraic equations. The direct interaction approximation (DIA) loses realisability above a critical Reynolds number; for p > 0, A > 0, it has a branch of spurious realisable self-excited solutions which subsist at zero driving. The renormalised vertex equations have no self-excited solution and a larger (but still finite) domain of realisability. In the quadratic DIA, obtained from an equivalent quadratic system for U and x = U*, both pathologies disappear.

Approximate representations for the SU(2) ordered exponential U(t‖E) =(exp[i∫t0dt’ σ⋅E(t’)])+, written as a functional of its input field E(t), are derived in the adiabatic ( ρ≪1) and stochastic ( ρ≫1) limits, where ρ≡‖dE/dt‖/E, E=E/E, E=+(E2)1/2. An algorithm is set up for the adiabatic case, and fixed‐point equations are obtained for situations of possible convergence. In the stochastic regime, ‘‘averaged’’ functions describing U(t‖E) are derived which reproduce its slowly varying dependence of large magnitude while missing, or approximating, rapid oscillations of small magnitude. Several functional integrals, analytic and machine are carried out over these approximate forms, and their results compared with the same functional integrals over the exact U(t‖E).

The critical behaviour of the Schlogl model is studied using field theoretical methods. It is shown that for the equal time correlation functions this behaviour is that of the well-known g ϕ 4 model.

A method of approximating an SU(2) ordered exponential U(t‖E), defined in terms of a two-dimensional input vector E (t)= E (t)E(t), E=+( E ) 2 1 2 , is suggested for the stochastic limit of strongly varying E, where ‖dE/ dt‖ ⪢ E. In the context of an “averaged” representation, where the high frequency fluctuations normally superimposed upon a relatively slowly-varying background are suppressed, U(t‖ (E) is given as a functional of E(t), suitable for use as an integrand in a functional integral with gaussian white-noise weighting.

Direct numerical simulations of decaying high Reynolds number turbulence are presented at a resolution of 20482 for a periodic flow with large scale symmetries. We characterize a transition of the inertial energy spectrum exponent from n ≃- 4 at early times to n ≃ -3 at latter times. In physical space, the first regime is associated with isolated vorticity gradient sheets, as predicted by Saffman (1971). The second regime corresponds to an enstrophy cascade (Kraichnan 1967, Batchelor 1969) and reflects the formation of layers resulting from the packing of vorticity gradient sheets.

It has early been realized that very high resolutions are required to simulate large Reynolds number turbulence, by direct numerical integration of the Navier-Stokes equation $$\begin{array}{*{20}{c}}{{\partial _t}V + V.\nabla V = - \nabla P + V\Delta V} \\ {\nabla .V = 0} \end{array}$$ (1) Focussing on the internal dynamics rather than on the effects of special geometries, we assume periodic boundary conditions. We thus use spectral methods for the space variables. We concentrate here on the simulation of a turbulent flow at a resolution of (2048)2 collocation points, and especially on visualizations of the small scales of the flow. Computations at lower resolutions have been reported previously[1].

The three-dimensional stability of saturated two-dimensional vortical states of planar mixing layers and jets is studied by direct integration of the Navier-Stokes equations. Small-scale instabilities are shown to exist for spanwise scales at which classical linear modes are stable. These modes grow on convective time scales. extract their energy from the mean flow. and persist to moderately low Reynolds numbers. Their growth rates are comparable to the most rapidly growing inviscid instability and to the growth rates of two-dimensional subharmonic (pairing) modes. The three-dimensional modes do not appear to saturate in quasi-steady states. Indeed. they seem to lead directly to chaos. Results are presented for the resulting three-dimensional turbulent states.

Three-dimensional (3D) superflows past a circular cylinder are studied by numerically integrating the nonlinear Schrodinger equation. 3D initial data are built from the two-dimensional (2D) stationary vortex nucleation solutions. Quasistationary half-ring vortices, pinned at the sides of the cylinder, are generated after a short time. On a longer time scale, either 3D vortex stretching induces dissipation and drag, or the vortex is absorbed by the cylinder. The corresponding 3D critical velocity is found to be well below the 2D one. The implications for experiments in Bose-Einstein condensed gas and low-temperature helium are discussed.

Non-linear waves described by the defocusing non-linear Schroedinger (NLS) equation admit a hydrodynamical representation in terms of Galilean potential flows and, using this correspondence, an autonomous equation for potential flow's non-linear acoustic has been recently derived by Nore et al. However, this equation does not contain simple solutions of the original one such as (dark) solitons. The purpose of the present article is to characterize the reasons behind this failure and to present an original method to build separate equations describing all different types of acoustic solutions (but one). For reasons of generality, we work in a framework adapted to special relativistic hydrodynamics. All the results we derive have Galilean counterparts which are also discussed. In particular, we argue that there exist an infinity of different acoustic sectors for relativistic barotropic fluids, and we prove this result for fluids with a particularly simple equation of state. Solitons are naturally captured by our approach and a few explicit examples are worked out. Conserved quantities for the acoustic regime are also derived.

The three-dimensional stability of two-dimensional vortical states of planar mixing layers is studied by direct numerical integration of the Navier-Stokes equations. Small-scale instabilities are shown to exist for spanwise scales at which classical linear modes are stable. These modes grow on convective timescales, extract their energy from the mean flow and exist at moderately low Reynolds numbers. Their growth rates are comparable with the most rapidly growing inviscid instability and with the growth rates of two-dimensional subharmonic (pairing) modes. At high amplitudes, they can evolve into pairs of counter-rotating, streamwise vortices, connecting the primary spanwise vortices, which are very similar to the structures observed in laboratory experiments. The three-dimensional modes do not appear to saturate in quasi-steady states as do the purely two-dimensional fundamental and subharmonic modes in the absence of pairing. The subsequent evolution of the flow depends on the relative amplitudes of the pairing modes. Persistent pairings can inhibit three-dimensional instability and, hence, keep the flow predominantly two-dimensional. Conversely, suppression of the pairing process can drive the three-dimensional modes to more chaotic, turbulent-like states. An analysis of high-resolution simulations of fully turbulent mixing layers confirms the existence of rib-like structures and that their coherence depends strongly on the presence of the two-dimensional pairing modes.

The free decay of high Reynolds two dimensional turbulence is simulated by direct numerical intergration of the Navier-Stokes equations at a resolution of 1024 x 1024 with symmetric random initial conditions. The following scenario is observed : At early times, large scale straining generates quasi-rectilinear vorticity gradient sheets with thickness decaying exponentially in time until dissipation becomes relevant.In Fourier space, the energy spectrum displays a k−n - range with n − 4, in agreement with Saffman's theory. Close to the time of maximum enstrophy dissipation, we observe a transition to an n − 3 inertial range, consistent which the Batchelor-Kraichnan theory of enstrophy cascade. In this regime, vorticity gradients are distributed on convoluted secondary dissipative structures resulting from folding and reconnection of early time sheets.

Stable localized wave packets in two‐dimensional incompressible Poiseuille flow are characterized by direct simulation of the Navier–Stokes equations. These asymmetric wave packets depend only on the Reynolds number when the channel periodicity length is sufficiently large. They are created through a saddle‐node bifurcation at Re ∼2330, well below the critical Reynolds number for finite‐amplitude Tollmien–Schlichting waves. Pairs of wave packets are found to interact repulsively, and no evidence for locking has been observed. Finally, an amplitude‐equation model is suggested which incorporates the spatial asymmetry of the localized solutions.

The dynamics of streamwise vorticity in axisymmetric jets is studied by direct numerical integration of the Navier–Stokes equations coupled with a passive scalar. Consistently with recent experiments and inviscid numerical simulations, the present viscous simulations show the appearance of pairs of axially counter‐rotating vortex filaments. After their formation the filaments move away from the jet, dragging the tracer into finger‐shaped structures. The three‐dimensional topology of the rings, filaments, and fingers is described, well beyond the time of filament formation. Finally, visualizations of the pressure gradient field are presented, which suggest that the filaments can be directly observed experimentally by seeding the flow with microbubbles.

The dynamics of both the inviscid and viscous Taylor–Green (TG) three-dimensional vortex flows are investigated. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier–Stokes equations (with up to 256 3 modes) and by power-series analysis in time. The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation in the form of vortex sheets located near the walls of this box. The flow is smooth but has complex-space singularities within a distance $\hat{\delta}(t)$ of real (physical) space which give rise to an exponential tail in the energy spectrum. It is found that $\hat{\delta}(t)$ decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place at later times, possibly leading to a real singularity ( $\hat{\delta}(t) = 0$ ) at a finite time. These direct integration results are consistent with new temporal power-series results that extend the Morf, Orszag & Frisch (1980) analysis from order t 44 to order t 80 . Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis. The viscous dynamics (decay) have been studied for Reynolds numbers R (based on an integral scale) up to 3000 and beyond the time t max at which the maximum energy dissipation is achieved. Early-time, high- R dynamics are essentially inviscid and laminar. The inviscidly formed vortex sheets are observed to roll up and are then subject to instabilities accompanied by reconnection processes which make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Near t max the small scales of the flow are nearly isotropic provided that R [gsim ] 1000. Various features characteristic of fully developed turbulence are observed near t max when R = 3000 and R λ = 110: a k − n inertial range in the energy spectrum is obtained with n ≈ 1.6–2.2 (in contrast with a much steeper spectrum at earlier times); th energy dissipation has considerable spatial intermittency; its spectrum has a k −1+μ inertial range with the codimension μ ≈ 0.3−0.7. Skewness and flatness results are also presented.

Direct numerical simulations of decaying high-Reynolds-number turbulence are presented at resolutions up to 8002 for general periodic flows and 20482 for periodic flows with large-scale symmetries. For turbulence initially excited at large scales, we characterize a transition of the inertial energy-spectrum exponent from n ≈ − 4 at early times to n ≈ − 3 when the turbulence becomes more mature. In physical space, the first regime is associated with isolated vorticity-gradient sheets, as predicted by Saffman (1971). The second regime, which is essentially statistical, corresponds to an enstrophy cascade (Kraichnan 1967; Batchelor 1969) and reflects the formation of layers resulting from the packing of vorticity-gradient sheets. In addition to these small-scale structures, the simulation displays vorticity macro-eddies which will survive long after the vorticity-gradient layers have been dissipated (McWilliams 1984). We validate the linear description of two-dimensional turbulence suggested by Weiss (1981), which predicts that coherent vortices will survive in regions where vorticity dominates strain, while vorticity-gradient sheets will be formed in regions where strain dominates. We show that this analysis remains valid even after vorticity-gradient sheets have been formed.

In this thesis several different Fourier Galerkin-truncated conservative systems are studied. It is shown that, in a very general way, these systems relax toward the thermodynamic equilibrium with a small-scale thermalization that induces an effective dissipation at large scales while conserving the global invariants. The first part of this work is concerned with the study of the effective viscosity of the truncated Euler equation, making use of the EDQNM closure theory and Monte-Carlo methods. We propose a two-fluid model of the system and this work is extended to the case of helical flows. The relaxation dynamics described by the two-dimensional truncated magnetohydrodynamics equations and three-dimensional compressible fluids is then characterized. In a second part, a generalization of the previous study to the truncated Gross-Pitaevski equation is given. Finite-temperature effects that are present in superfluids, such as mutual friction and couterflow, are found to be naturally included in the truncated Gross-Pitaevski equation. This system thus appears as simple and rich model of superfluidity at finite temperature. Finally, the radiation produced by moving point Gross-Pitaevskii vortices is studied analytically and numerically., Cette thèse regroupe des études portant sur la dynamique de relaxation de différents systèmes conservatifs ayant tous une troncature de Galerkin sur les modes de Fourier. On montre que, de façon très générale, ces systèmes relaxent lentement vers l'équilibre thermodynamique avec une thermalisation partielle à petite échelle qui induit une dissipation effective à grande échelle, tout en conservant les invariants globaux. La première partie de ce travail est consacrée à l'étude de la viscosité effective dans l'équation d'Euler incompressible tronquée. L'utilisation des méthodes de Monte-Carlo et de la théorie EDQNM permet la construction d'un modèle à deux fluides de ce système. Cette étude est ensuite généralisée au cas des écoulements hélicitaires. La dynamique de relaxation des écoulements décrits par les équations de la magnétohydrodynamique et des fluides compressibles tronqués est finalement caractérisée. Dans une deuxième partie, nous généralisons l'étude de la thermalisation au cas de l'équation de Gross-Pitaevski tronquée. On trouve que des effets existant dans les superfluides à température finie, comme la friction mutuelle et le ''counterflow'', sont naturellement présents dans ce modèle. On propose ainsi l'équation de Gross-Pitaevskii tronquée comme un modèle simple et riche de la dynamique superfluide à température finie. La radiation produite par le mouvement de vortex ponctuels décrits par l'équation de Gross-Pitevskii 2D est finalement caractérisée analytiquement et numériquement.